For the reconstruction problems, we used accelerated proximal gradient descent [1]}, as summarized in Alg. REF  with \( J = {1e3} \) , \( \gamma _1 = 0.5 \) , \( \gamma _2 = 1.5^{-1} \) .
We solve the proximal operator \( \operatorname{prox} \colon \mathbb {R}^n \rightarrow \mathbb {R}^n \) , which for \( H \colon \mathbb {R}^n \rightarrow \mathbb {R}\)  and \( \tau \in \mathbb {R}^+ \)  is defined as
\(\operatorname{prox}_{\tau H} (y) = \operatornamewithlimits{arg\,min}_{x} \tau H(x) + \frac{1}{2} \left\Vert x - y\right\Vert _2^2\) 
